Eray Ozkural wrote:
>I believe this is an interesting point. So why do physicists think
of R as an adequate model for "real" space geometry
at all? Is it just an accident of history or do they seriously believe
that unmeasurable things exist?
I can only speak for myself, but as a physicist, I have a couple
of criteria for what kinds of pathological objects, such as unmeasurable
sets, are unlikely to be interesting physically. (1) The kinds of
objects involved in the Banach-Tarski paradox are clearly not
interesting physically, because they owe their existence to a
historical accident. If Newton and Leibniz's ideas had been
formalized using smooth infinitesimal analysis instead of sets
of points in R^n, then such objects would never have come up.
(2) The correspondence principle tells us that we should expect
any physical theory to be superseded later on by a more general
theory. Therefore, any treatment of infinities or infinitesimals
is really describing a limiting process that has to stop when we
get beyond the frontiers of the relevant theory. For example, when
a physicist says there's a discontinuity in density at the surface
of a lake, he's talking about a limiting process that he knows
will stop when he reaches the scale of individual atoms. He expects
this to happen with *all* limiting processes, even if he doesn't
know the nature of the more general theory or exactly at what
point the breakdown will happen.